Asymptotic Normality of M-Estimators

This section sets forth conditions for the asymptotic normality ofM-estimators in addition to the conditions for consistency An estimator в of a parameter в0 є Km is asymptotically normally distributed if an increasing sequence of positive numbers an and a positive semidefinite m x m matrix £ exist such that an(в — в0) ^d Nm [0, £]. Usually, an = – Jn, but there are exceptions to this rule.

Asymptotic normality is fundamental for econometrics. Most of the econo­metric tests rely on it. Moreover, the proof of the asymptotic normality theorem in this section also nicely illustrates the usefulness of the main results in this chapter.

Given that the data are a random sample, we only need a few additional conditions over those of Theorems 6.10 and 6.11:

Theorem 6.1: Let, in addition to the conditions of Theorems 6.10 and 6.11, the following conditions be satisfied:

where Q"(в) is the second derivative of Q^) = E[g(X1, в)]. Then it follows from (6.39)-(6.41) and Theorem 6.12 that

plim Q"(вс + 1(0 — вс)) = Q"(вс) = 0. (6.42)

n^TO

Note that Q"(во) corresponds to the matrix A in condition (e), and thus Q"(во) is positive in the “argmin” case and negative in the “argmax” case. Therefore, it follows from (6.42) and Slutsky’s theorem (Theorem 6.3) that

plim Q"(в0 + 1(0 — в0))—1 = Q//(в0)—1 = A—1. (6.43)

n^TO

Now (6.38) can be rewritten as

– во) = -Q"(во + i(0 – 0o))-1VnQ'(во)

+ Q"(во + к(в – во))-1^'(в)

= – Q"(во + і(в – во)УХ4Пй'(во) + Op(1), (6.44)

where the op(1) term follows from (6.37), (6.43), and Slutsky’s theorem. Because of condition (b), the first-order condition for во applies, that is,

Q'(во) = E[dg(X1, во)/dво] = о. (6.45)

Moreover, condition (f), adapted to the univariate case, now reads as follows:

var[dg(Xь в^/dво] = B e (о, то). (6.46)

Therefore, it follows from (6.45), (6.46), and the central limit theorem (Theorem 6.23) that

n

VnQ ‘(во) = (1/Vn)J2 dg(Xj, во)/dво ^d N [о, B]. (6.47)

j=1

Now it follows from (6.43), (6.47), and Theorem 6.21 that

– Q "(во + і(в – во)Ух^пй ‘(во) ^d N [о, A-1BA-1]; (6.48)

hence, the result of the theorem under review for the case m = 1 follows from (6.44), (6.48), and Theorem 6.21. Q. E.D.

The result of Theorem 6.28 is only useful if we are able to estimate the asymptotic variance matrix A-1BA-1 consistently because then we will be able to design tests of various hypotheses about the parameter vector во.